TITLE: Finding the value of $\sin {\frac{31 \pi}3}$
QUESTION [6 upvotes]: The task was to find out the value of $$\sin\frac{31\pi}3$$
This is a example in my book in which following steps are shown:
$$\begin{align}
\sin\frac{31\pi}3& =\sin\left(10\pi +\frac\pi3\right)\\
&=\sin\frac\pi3\\
&=\frac{\sqrt3}2
\end{align}$$
I cannot understand step 3, $=\sin\frac\pi3$

REPLY [4 votes]: Think of the unit circle and the definition of $\sin$: A straight line drawn from the origin with the angle between the $x$-axis of $\theta$ intersects the unit circle at $(\cos \theta, \sin \theta)$. 

Say you have a point on the unit circle at $(\cos \theta, \sin \theta)$. If the line making the angle $\theta$ from the origin makes a full revolution (revolves around the origin by a measure of $2\pi$), then it will end up in the same position. Thus, $\sin n = \sin (2\pi + n) = \sin (n - 2\pi)$.